2 research outputs found
Identifying codes in line digraphs
Given an integer , a -identifying code in a digraph
is a dominating subset of vertices such that all distinct subsets of
vertices of cardinality at most have distinct closed in-neighbourhood
within . In this paper, we prove that every -iterated line digraph of
minimum in-degree at least 2 and , or minimum in-degree at least 3 and
, admits a -identifying code with , and in any
case it does not admit a -identifying code for .
Moreover, we find that the identifying number of a line digraph is lower
bounded by the size of the original digraph minus its order. Furthermore, this
lower bound is attained for oriented graphs of minimum in-degree at least 2
Identifying codes in line digraphs
Given an integer ≥1, a (1, ≤ )-identifying code in a digraph is a dominating subset C of vertices such that all distinct subsets of vertices of cardinality at most have distinct closed in-neighborhoods within C . In this paper, we prove that every line digraph of min- imum in-degree one does not admit a (1, ≤ )-identifying code for ≥3. Then we give a characterization so that a line digraph of a digraph different from a directed cycle of length 4 and minimum in-degree one admits a (1, ≤2)-identifying code. The identifying number of a digraph D , −→ γID (D ) , is the minimum size of all the identifying codes of D . We establish for digraphs without digons with both vertices of in-degree one that −→ γID (LD ) is lower bounded by the number of arcs of D minus the number of vertices with out-degree at least one. Then we show that −→ γID (LD ) attains the equality for a digraph having a 1- factor with minimum in-degree two and without digons with both vertices of in-degree two. We finish by giving an algorithm to construct identifying codes in oriented digraphs with minimum in-degree at least two and minimum out-degree at least one.This research is supported by MICINN from the Spanish Government under project PGC2018-095471-B-I00 and partially by AGAUR from the Catalan Government under project 2017SGR1087 . The research of the second author has also been supported by MICINN from the Spanish Government under project MTM2017-83271-R. The second and third authors have received funding research and innovation programme under the Marie Sklodowska-Curie grant agree- ment No 734922